Linear Algebra

Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$

Ohio State University exam problems and solutions in mathematics

Problem 699

(a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix}
2 & 0 & -1 \\
0 &2 &-1 \\
-1 & 0 & 1
\end{bmatrix}.\]

(b) Find the inverse matrix of $A$.

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Solution

(a) Find a $3\times 3$ nonsingular matrix $A$.

Assume that $A$ is nonsingular. Then the inverse matrix $A^{-1}$ exists.
Multiplying the given equality by $A^{-1}$ on the left, we obtain
\[3A^{-1}A=A^{-1}(A^2+AB)=A^{-1}A^2+A^{-1}AB=A+IB=A+B.\] Note that the left most term is equal to $3I$.
Hence, we have $3I=A+B$. Solving for $A$, we have
\[A=3I-B=\begin{bmatrix}
3 & 0 & 0 \\
0 &3 &0 \\
0 & 0 & 3
\end{bmatrix}-\begin{bmatrix}
2 & 0 & -1 \\
0 &2 &-1 \\
-1 & 0 & 1
\end{bmatrix}=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &1 \\
1 & 0 & 2
\end{bmatrix}.\]

As we see in part (b), this matrix is actually invertible.

(b) Find the inverse matrix of $A$.

To find the inverse matrix of $A$, we reduce the augmented matrix $[A\mid I]$ as follows:
\begin{align*}
[A\mid I]= \left[\begin{array}{rrr|rrr}
1 & 0 & 1 & 1 &0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 2 & 0 & 0 & 1 \\
\end{array} \right] \xrightarrow{R_3-R_2}
\left[\begin{array}{rrr|rrr}
1 & 0 & 1 & 1 &0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 & 1 \\
\end{array} \right]\\[6pt] \xrightarrow[R_2-R_3]{R_1-R_3}
\left[\begin{array}{rrr|rrr}
1 & 0 & 0 & 2 &0 & -1 \\
0 & 1 & 0 & 1 & 1 & -1 \\
0 & 0 & 1 & -1 & 0 & 1 \\
\end{array} \right].
\end{align*}
The left 3 by 3 part is now the identity matrix.
So the inverse matrix is given by the right 3 by 3 part:
\[A^{-1}=\begin{bmatrix}
2 & 0 & -1 \\
1 &1 &-1 \\
-1 & 0 & 1
\end{bmatrix}.\]

Common Mistake

This is a midterm exam problem of Lienar Algebra at the Ohio State University.

One common mistake is: $A=3-B$. Don’t forget the identity matrix $I$. $3$ is a number but $3I$ is a matrix.
Also $3I$ is the 3 by 3 matrix whose diagonal entries are 3 and 0 elsewhere. Note that $3I$ is not the 3 by3 matrix whose entries are all 3.

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